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Nonlinear Evolution Equations of the Soliton Type : Old and New Results
No full textAn overview on the study of nonlinear evolution equations of soliton type is provided. In addition, 5th-order nonlinear evolution equations are shown to be connected to the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation via Bäcklund transformations. The links are depicted in a wide net of links which we term a Bäcklund Chart. The links obtained previously by Rogers and Carillo and by Carillo and Fuchssteiner are revisited, and new results are obtained. A 5th-order nonlinear evolution equation, which does not seem to appear in any list of integrable equations, is provided. All the connected equations exhibit a very interesting symmetry structure enjoyed by the corresponding full hierarchies. Indeed, they all admit a hereditary recursion operator. Hence, each one of the mentioned equations represents the base member of a corresponding hierarchy of equations. These hierarchies are constructed via the recursive application of the respective recursion operators. The symmetry properties of such equations are recalled. Finally, we compare the net of links, derived via Bäcklund transformations, in the case of the fifth-order nonlinear evolution equations with an analog net of links connecting third-order Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. Analogies and discrepancies between the connections established in the case of fifth-order equations with respect to those established in the case of third-order equations are analyzed. This study aims to open the way for the construction of corresponding non-Abelian equations of the fifth order
Abelian versus non-abelian baecklund charts:some remarks
Full text linkConnections via B ̈acklund transformations among different non- linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via B ̈acklund transfor- mations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities be- tween Ba ̈cklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The B ̈acklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non- commutative cases, are richer than the corresponding commutative ones
Soliton equations: admitted solutions and invariances via B\"acklund transformations
Full text linkA couple of applications of B\"acklund transformations in the study of
nonlinear evolution equations is here given. Specifically, we are concerned
about third order nonlinear evolution equations. Our attention is focussed on
one side, on proving a new invariance admitted by a third order nonlinear
evolution equation and, on the other one, on the construction of solutions.
Indeed, via B\"acklund transformations, a {\it B\"acklund chart}, connecting
Abelian as well as non Abelian equations can be constructed. The importance of
such a net of links is twofold since it indicates invariances as well as allows
to construct solutions admitted by the nonlinear evolution equations it
relates. The present study refers to third order nonlinear evolution equations
of KdV type. On the basis of the Abelian
wide B\"acklund chart which connects various different third order nonlinear
evolution equations an
invariance admitted by the {\it Korteweg-deVries interacting soliton}
(int.sol.KdV) equation is obtained and a related new explicit solution is
constructed.
Then, the corresponding non-Abelian {\it B\"acklund chart}, shows how to
construct matrix solutions of the mKdV equations: some recently obtained
solutions are reconsidered.Comment: 11 pages, 6 figures. arXiv admin note: text overlap with
arXiv:2101.0924
Recursion operators admitted by non-Abelian Burgers equations. Some remarks
No full textThe recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here considered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as rt = rxx + 2rxr. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be hereditary. These results are proved via direct computations as well as via computer assisted manipulations; ad hoc routines are needed to treat non-Abelian quantities and relations among them. The obtained recursion operator generates the mirror non-Abelian Burgers hierarchy. The latter, when the unknown operator r is replaced by a real valued function reduces to the usual (commutative) Burgers hierarchy. Accordingly, also the recursion operator reduces to the usual Burgers one
Baecklund transformations and non abelian nonlinear evolution equations. A novel Baecklund chart
No full textThird order Non Abelian Nonlinear Evolution Equations, such as potential Ko- rteweg deVries (pKdV), Korteweg deVries (KdV) as well as different versions of modified Korteweg deVries (mKdV), and the Korteweg deVries interacting soliton and Korteweg de- Vries singuarity manifold equations are here considered. In particular a new B ̈acklund Chart is constructed. Notably it, on one side, generalizes the Abelian B ̈acklund Chart presented in [24] and, on the other one, extends the noncommutative B ̈acklund Chart obtained in [12] to incorporate further non Abelian operator equations. The Ba ̈cklund transformations which link the various nonlinear evolution equations, allow to write the corresponding recursion operators and, hence, the corresponding whole hierarchies of nonlinear operators equations are connected
A novel noncommutative KdV-type equation, its recursion operator, and solitons
No full textA noncommutative KdV-type equation is introduced extending the Bäcklund chart in Carillo et al. [Symmetry Integrability Geom.: Methods Appl. 12, 087 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in Olver and Sokolov [Commun. Math. Phys. 193, 245 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies, and an explicit solution class are derived
Non Abelian Nonlinear Evolution Equations & Baecklund Transformations
No full textStructural Properties of Non Abelian Nonlinear Evolution Equationsare studied. Specifically, third order Non Abelian Nonlinear Evolution Equations, such as potential Korteweg deVries (pKdV), Korteweg deVries (KdV) as well as different versions of modified
Korteweg deVries (mKdV), and the Korteweg deVries interacting soliton and Korteweg
deVries singuarity manifold equations and links among them are studied [1,2,3]. The
Bäcklund Transformations connecting all these equations are depicted in aBäcklund
Chart. It represents an extension of the Bäcklund Chart in [4] obtained in the case
of nonlinear evolution equations in an unknown real valued function. Also in the non
Abelian case all the studied equations are proved to admit a hereditary recursion operator. Hence the corresponding hierarchies are constructed. Results on non Abelian
Burgers equations and hierarchies are also mentioned [5,6]
B\"acklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations
Full text linkThe KdV eigenfunction equation is considered: some explicit solutions are
constructed. These, to the best of the authors' knowledge, new solutions
represent an example of the powerfulness of the method devised. Specifically,
B\"acklund transformation are applied to reveal algebraic properties enjoyed by
nonlinear evolution equations they connect. Indeed, B\"acklund transformations,
well known to represent a key tool in the study of nonlinear evolution
equations, are shown to allow the construction of a net of nonlinear links,
termed "B\"acklund chart", connecting Abelian as well as non Abelian equations.
The present study concerns third order nonlinear evolution equations which are
all connected to the KdV equation. In particular, the Abelian wide B\"acklund
chart connecting these nonlinear evolution equations is recalled. Then, the
links, originally established in the case of Abelian equations, are shown to
conserve their validity when non Abelian counterparts are considered. In
addition, the non-commutative case reveals a richer structure related to the
multiplicity of non-Abelian equations which correspond to the same Abelian one.
Reduction from the nc to the commutative case allow to show the connection of
the KdV equation with KdV eigenfunction equation, in the "scalar" case.
Finally, recently obtained matrix solutions of the mKdV equations are
recalled.Comment: 14 pages, 6 figures, conference FASNET 2020 (online
Construction of soliton solutions of the matrix modified Korteweg-de Vries equation
Full text linkAn explicit solution formula for the matrix modified KdV equation is
presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo
Schiavo, and C. Schiebold. Matrix solitons solutions of the modified
Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and
Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado,
G. Stepan (Springer, Cham, 2020), pp. 75-83). In fact, the solutions in Ref.7
are part of a subclass studied in detail by the authors in a forthcoming
publication. Here several solutions beyond this subclass are constructed and
discussed with respect to qualitative properties.Comment: 10 pages, 6 figures, Proceedings of the Second International
Nonlinear Dynamics Conference (NODYCON 2021), W. Lacarbonara et al, Ed.
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